Seminar: Brane quantization and the double affine Hecke algebra

Double affine Hecke algebras, first introduced by Ivan Cherednik, are a family of algebras labeled by a choice of root system. Cherednik used their representation theory to give a uniform proof of the Macdonald constant-term conjectures; thereafter, they have been a source of interest in mathematics and also physics, especially after they were used to construct homological invariants of torus knots by Gorsky, Oblomkov, Rasmussen, and Shende. I'll discuss some new calculations that recover aspects of DAHA representation theory from the study of a particular Hitchin moduli space, in the framework of the brane quantization program of Gukov and Witten. In the associated physical picture, the so-called "spherical" DAHA appears as an algebra of line operators in four-dimensional N = 2* theory. This is ongoing work, to appear soon with Sergei Gukov, Peter Koroteev, and Satoshi Nawata.